June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or rise in a certain base. Take this, for example, let's say a country's population doubles annually. This population growth can be depicted in the form of an exponential function.

Exponential functions have multiple real-life uses. In mathematical terms, an exponential function is displayed as f(x) = b^x.

Here we will learn the basics of an exponential function coupled with relevant examples.

What’s the equation for an Exponential Function?

The generic formula for an exponential function is f(x) = b^x, where:

b is the base, and x is the exponent or power.
b is a constant, and x varies

As an illustration, if b = 2, we then get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.

In cases where b is greater than 0 and does not equal 1, x will be a real number.

How do you chart Exponential Functions?

To chart an exponential function, we must discover the spots where the function crosses the axes. These are called the x and y-intercepts.

Considering the fact that the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.

To discover the y-coordinates, we need to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

In following this approach, we determine the range values and the domain for the function. Once we determine the rate, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical qualities. When the base of an exponential function is larger than 1, the graph is going to have the below characteristics:

The line intersects the point (0,1)
The domain is all positive real numbers
The range is larger than 0
The graph is a curved line
The graph is increasing
The graph is smooth and continuous
As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
As x nears positive infinity, the graph rises without bound.

In instances where the bases are fractions or decimals between 0 and 1, an exponential function displays the following properties:

The graph intersects the point (0,1)
The range is more than 0
The domain is all real numbers
The graph is descending
The graph is a curved line
As x nears positive infinity, the line in the graph is asymptotic to the x-axis.
As x approaches negative infinity, the line approaches without bound
The graph is smooth
The graph is constant

Rules

There are several essential rules to remember when working with exponential functions.

Rule 1: Multiply exponential functions with an equivalent base, add the exponents.

For example, if we need to multiply two exponential functions with a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with an equivalent base, subtract the exponents.

For instance, if we need to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y).

Rule 3: To increase an exponential function to a power, multiply the exponents.

For instance, if we have to grow an exponential function with a base of 4 to the third power, we are able to compose it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is always equivalent to 1.

For instance, 1^x = 1 no matter what the value of x is.

Rule 5: An exponential function with a base of 0 is always equivalent to 0.

For example, 0^x = 0 despite whatever the value of x is.

Examples

Exponential functions are commonly used to signify exponential growth. As the variable grows, the value of the function increases quicker and quicker.

Example 1

Let’s examine the example of the growing of bacteria. If we have a culture of bacteria that multiples by two every hour, then at the close of hour one, we will have twice as many bacteria.

At the end of hour two, we will have 4 times as many bacteria (2 x 2).

At the end of hour three, we will have 8x as many bacteria (2 x 2 x 2).

This rate of growth can be represented an exponential function as follows:

f(t) = 2^t

where f(t) is the total sum of bacteria at time t and t is measured hourly.

Example 2

Moreover, exponential functions can portray exponential decay. Let’s say we had a radioactive substance that decomposes at a rate of half its quantity every hour, then at the end of one hour, we will have half as much material.

At the end of two hours, we will have a quarter as much substance (1/2 x 1/2).

At the end of three hours, we will have 1/8 as much material (1/2 x 1/2 x 1/2).

This can be displayed using an exponential equation as below:

f(t) = 1/2^t

where f(t) is the amount of material at time t and t is assessed in hours.

As shown, both of these examples use a similar pattern, which is the reason they can be depicted using exponential functions.

In fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base remains fixed. This indicates that any exponential growth or decay where the base varies is not an exponential function.

For example, in the matter of compound interest, the interest rate continues to be the same while the base is static in ordinary amounts of time.

Solution

An exponential function can be graphed using a table of values. To get the graph of an exponential function, we need to input different values for x and measure the corresponding values for y.

Let us check out the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

To start, let's make a table of values.

As demonstrated, the rates of y increase very rapidly as x increases. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like this:

As shown, the graph is a curved line that rises from left to right and gets steeper as it persists.

Example 2

Draw the following exponential function:

y = 1/2^x

First, let's make a table of values.

As shown, the values of y decrease very rapidly as x increases. This is because 1/2 is less than 1.

Let’s say we were to chart the x-values and y-values on a coordinate plane, it is going to look like what you see below:

This is a decay function. As you can see, the graph is a curved line that gets lower from right to left and gets smoother as it proceeds.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions exhibit particular characteristics where the derivative of the function is the function itself.

The above can be written as following: f'x = a^x = f(x).

Exponential Series

The exponential series is a power series whose expressions are the powers of an independent variable figure. The common form of an exponential series is:

Source

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